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Stochastic Gradient Descent (SGD) is a widely used, foundational algorithm in data science and machine learning. As a result, analyses of SGD abound making use of a variety of assumptions, especially on the noise behavior of the stochastic gradients. While recent works have achieved a high-degree of generality on assumptions about the noise behavior of the stochastic gradients, it is unclear that such generality is necessary. In this work, we construct a simple example that shows that less general assumptions will be violated, while the most general assumptions will hold.
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Global Convergence and Stability of Stochastic Gradient Descent
In machine learning, stochastic gradient descent (SGD) is widely deployed to train models using highly non-convex objectives with equally complex noise models. Unfortunately, SGD theory often makes restrictive assumptions that fail to capture the non-convexity of real problems, and almost entirely ignore the complex noise models that exist in practice. In this work, we demonstrate the restrictiveness of these assumptions using three canonical models in machine learning. Then, we develop novel theory to address this shortcoming in two ways. First, we establish that SGD’s iterates will either globally converge to a stationary point or diverge under nearly arbitrary nonconvexity and noise models. Under a slightly more restrictive assumption on the joint behavior of the non-convexity and noise model that generalizes current assumptions in the literature, we show that the objective function cannot diverge, even if the iterates diverge. As a consequence of our results, SGD can be applied to a greater range of stochastic optimization problems with confidence about its global convergence behavior and stability.
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- Award ID(s):
- 2023239
- PAR ID:
- 10388897
- Date Published:
- Journal Name:
- Conference on Neural Information Processing Systems
- Volume:
- 36
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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